A TAXONOMY OF AUTOMATED THEOREM PROVING ENVIRONMENTS

GORDON BEAVERSUniversity of the Ozarks

HAL BERGHELUniversity of Arkansas

Abstract

In this paper we (a) draw attention to the role that normal forms play in
various Automated Theorem Proving (ATP) procedures and (b) introduce analogous
ATP procedures with complementary normal forms. Since this role is most easily
examined in propositional logic, our discussion will be so limited. However,
the dual propositional ATP procedures generalize to first order logic in a
manner analogous to the way propositional resolution and propositional tableau
procedures do. We consider each of resolution, semantic tableaux and Wang's
algorithm as a consequence relation following the example of Gentzen's sequent
calculus. These familiar procedures are the paradigm ATP's depending on
specific normal form implementations and as such are suggestive of further
procedures.

INTRODUCTION

First a definition for well-formedness of our object language is provided.
The vocabulary for propositional logic consists of the following:

Where ' º ' is an element of {&,v,-,-}, and A,B,..., with or without
subscripts, define a countable infinitude of meta-logical variables which denote
arBitrary wffs. By convention, we will refer to wffs which contain no Binary
connectives as atomic.

Central to this discussion is the notion of a consequence relation. A
consequence relation is a set of ordered pairs of sets of wffs of

the language of propositional logic. The ordered pairs are regarded as
statements of the meta-language written in the form,

t |- d, where t and d are sets of wffs, possibly empty, and having the
interpretation d is a consequence of t. A consequence relation for
propositional logic has the following properties:

reflexivity A |- A for any formula A,

weakening t

d t |- d
---------- ----------
t,{A} |- d t |- d,{A}

transitivity (cut)

t1 |- d1,A A,t2|- d2
-----------------------
t1,t2 |- d1,d2

where the dashed line indicates that the consequence relation below follows
from the consequence relation(s) above.

As a notational convenience, we use expressions of the form "t,d"
in place of "t U d", and "t,A" in place of "t U {A}".

Perhaps the most fundamental interpretation of t |- d is given by the
model-theoretic semantics of the object language defined as follows. An
interpretation, I, is a mapping of a propositional variable onto one of the
truth values, true/false. Symbolically, we write I(p)=T for 'the interpretation
of p is true', for arBitrary propositional variable, p. Similarly, an n-ary
truth function is a mapping from an n-tuple of truth values onto a truth value.
The truth functions of immediate interest will be those which correspond to our
truth-functional operators. Specifically, let u,v is an element of {T,F}. The five truth
functions are:

f-(u) = F if u=T, and T otherwise

f&amp(u,v) = T if u=v=T, and F otherwise

fv(u,v) = F if u=v=F, and T otherwise

f-(u,v) = F if u=T and v=F, and T otherwise

f-(u,v) = T if u=v, and F otherwise

If we use these definitions together with the inductive definition of
well-formedness, we can see that interpretations of variables ultimately induce
a truth-value, called an evaluation, onto a wff.

We augment these definitions with the coincidence theorem for propositional
logic which ensures that evaluations of formulas are completely determined by
the interpretations of contained propositional variables. We note that the
operators -,&,v,-,-, discussed above correspond roughly to the
ordinary language analogs 'not', 'and', 'or', 'if...then...', and '...if and
only if...', respectively.

NORMAL FORMS

It is well-known that the set of truth functional operators

{-,&,v,-,-}, commonly referred to as the Principia Mathematica
[8] connectives, form a truth-functionally complete set. That is, any
truth-functional evaluation which can be expressed at all can be expressed by
means of some wff which contains only Principia Mathematica connectives.

However, with the uncontroversial definitional equivalences

(A-B) =df (A-B) & (B-A) , and

(A-B) =df (-AvB)

it becomes clear that this set is also truth-functionally redundant, for -
and - are eliminable without loss of truth-functional expressiveness. The
truth-functional independence of {-,&,v} is interesting primarily because it
makes it possible to express wffs in normal form.

There are two basic normal form representations of wffs in propositional
logic. A conjunctive normal form (CNF) is a conjunction of the form A1 & A2
& ... & An where each Ai, 1<=i<=n, is an elementary disjunction
of the form B1 v B2 v ,...,v Bm, m>=1, where each Bj, 1<=j<=m, is
atomic. Conversely, a disjunctive normal form (DNF) is a disjunction of
elementary conjuncts. It is well-known that for any arBitrary wff, A, there
exist formulas in both CNF and DNF which are truth-functionally equivalent to
it. That is, e(A) = e(CNF(A)) = e(DNF(A)), for all wffs, A. DeMorgan's laws
together with the associative, commutative and distributive laws for & and
v, familiarity of which is assumed, are adequate to illustrate this point.

For simplicity of exposition, we also introduce a third normal form,
negation normal form (NNF), which is defined inductively as follows:

(i) A conjunction or disjunction of atomic formulas is in NNF

(ii) if A and B are in NNF then so is A ± B

where ± is an element of {&,v}. It is easy to see that every wff has an NNF
equivalent. We return to the notion of normal forms in the next section.

MODEL-THEORETIC SEMANTICS

The inductive definitions for wff and wff-evaluation described above provide
the foundation of a model-theoretic semantics for propositional logic. Consider
an interpretation, I, of propositional variables and a corresponding evaluation,
e, induced by I onto some formula A. If e(A) = t under I, I is said to be a
model of A. It follows that for an arBitrary interpretation, I,

i. I is a model of p if and only if I(p) = t

ii. I is a model of (-A) if and only if I is not a model of A

iii. I is a model of (A º B) if and only if

(I is a model of A) ? (I is a model of B)

where ? is the metalanguage equivalent of º. If there exists at least
one interpretation which is a model of a formula, the formula is said to be
satisfiable. Following convention, we say that if all (no) interpretations are
models of a formula, the formula is tautologous (contradictory).

We can generalize the notion of model to a set of formulas, t, without
difficulty. We will say that any interpretation is a model of a set of formulas
just in case it is a model of each of the formulas individually. If there
exists at least one interpretation which is a model of t, t is said to be
satisfiable.

We further extend the analysis to the level of argument form. Let t be a
set of formulas and A be an individual formula. If for all interpretations, I,
which are models of t are also models of A, a valid consequence relation is said
to hold between t and A. By convention we refer to the formulas of t as
premises and to A as the conclusion of an argument form designated by t|-A.

As mentioned above, any wff can be transformed into an equivalent formula in
either CNF or DNF. It is appropriate that we here introduce two observations
concerning model-theoretic properties of normalized expressions:

(1) Any formula in CNF is tautologous iff each elementary disjunction
contains both a propositional variable and its negation.

and, conversely,

(2) Any formula in DNF is contradictory iff each elementary conjunction
contains both a propositional variable and its negation.

BINARY RESOLUTION

Binary resolution, in one of its myriad forms, is at the heart of most
modern inference engines and logic programming translators (e.g., Prolog,
Parlog,etc.). In its simplest form, Binary resolution relies upon a single rule
of inference - a generalization of the law of disjunctive syllogism (from
formulas of the form A v B and -A we may conclude B), the general form of which
is:

As is evident from the syntax, generalized disjunctive syllogism only
applies to wffs in CNF. Aside from its transparent validity, disjunctive
syllogism also has the advantage of excluding contradictory atomic formulas from
consideration.

There is a subset of CNF which is important from the point of view of the
automation of Binary resolution. This is the class of Horn clauses, or CNF
expressions in which the disjunctions contain at most one unnegated formula. In
terms of CNF, we may view Horn clauses as conjunctions of disjunctions of the
form Ai v -B1 v ... v -Bj, for non-negative i,j. We refer to the case when i=1
and j=0 as an assertion or fact. When i=1 and j>=1, we have a conditional
expression or rule. Goals are created when i=0 and j>=1. Finally, we
define the empty clause, , as the special case when i=j=0. Since there can be
no more than one unnegated disjunct, the set of Horn clauses is a proper subset
of CNF expressions.

We complement the notion of Horn clause with the notions of clause and dual
clause. By clause, we refer to the disjunction of atomic formulas (as in the
elementary disjuncts of CNF expressions, above). A dual clause, on the other
hand is a conjunction of atomic formulas.

While this limitation renders Binary resolution for Horn clauses essentially
incomplete for propositional logic, it offers considerable computational
advantage. From a procedural point of view, a single unnegated disjunct may
always be used as the 'entry point' into a clause set while all negated
disjuncts may be consistently interpreted as procedure calls. This simple
procedural interpretation allows a straightforward implementation of resolution.
It is important to recognize that resolution is defined for CNF expressions
generally, and not just for Horn clauses. However, the automation of resolution
is exceedingly difficult for non-Horn clause expressions.

An illustration of Binary resolution appears below. Consider the following
Horn clause:

By convention, we refer to each successive line as the resolvent of the
earlier line. Since the reduction was complete, i.e. produced the null clause,
, the Horn clause is not satisfiable. This is to say that there is no
interpretation under which the evaluation of the formula becomes True. This
technique may be easily extended to determine whether a consequence relation
exists between premises P1,...,Pk and conclusion, C, by testing the equivalent
expression A = P1 & ... & Pk & -C for satisfiability where A is in
CNF. Since for any consistent set of premises, P, and putative conclusion, C, P
|- C if and only if it is not the case that P |- -C, resolution is
amenable to indirect proof strategies.

Automated resolution was first suggested by RoBinson [4] and later
incorporated into the kernel of the high-level programming language Prolog by
Colmerauer and Kowalski. For further discussion, see [3].

THE SEMANTIC TABLEAU

The Semantic Tableau, or Truth Tree, is another approach to inferencing
which is convenient and, efficiency notwithstanding, relatively easy to automate
[1]. As with Binary resolution, the tableau is ideally suited for indirect
proof strategies whereby the premises of an argument form together with the
negation of the conclusion are tested for joint satisfiability. The tableau
rules of passage are tree-like in form (hence the nickname truth tree) as is
shown in the following rule schemata for CNF expressions:

As the algorithm makes clear, the lack of open paths after rules have been
applied to all remaining formulas indicates that the argument form is valid.

HILBERT OR FREGE SYSTEMS

Another standard consequence relation is defined by the axiomatic or Hilbert
style presentation of propositional logic. In this case t |- d just in
case for finite subsets t1 and d1 of t and d, respectively, &(t1) -
v(d1) is derivable from the axioms:

A - (B - A)

(A - (B - C)) - ((A - B) - (A - C))

(-B - -A) - (A - B)

using the rules modus ponens and substitution. The notation &(t1) and
v(d1) stand for the conjunction of the elements of t1 and the disjunction of
elements of d1.

The deduction and compactness theorems for propositional logic yield that t |-
d just in case |- &(t1) - v(d1). The standard soundness and
completeness results for classical logic show that axiomatic consequence and
model-theoretic consequence are identical. We observe that normal forms play no
role in Hilbert systems.

We now proceed to an abbreviated Gentzen style presentation of propositional
logic, that is, yet another way of presenting the notion of consequence relation
in propositional logic.

GENTZEN SYSTEMS

The Gentzen system for classical propositional logic has the structural
rules reflexivity, weakening and transitivity given above and logical rules of
inference for the connectives. Since we are considering only formulas in NNF we
present inference rules for only -, & and v, those rules being.

WANG'S ALGORITHM

Hao Wang implemented the following algorithm for propositional logic on an
IBM 704. He reported his results in Wang [7]. The algorithm essentially
delivers a cut-free proof using the & and v rules of Gentzen's sequent
calculus. We modify Wang's original rules to get:

Translate the formula into negation normal form. & on the left and v
on the right are replaced by commas,

Eliminate negations using the rules for negation in the Gentzen sequent
calculus given above.

eliminate & on the right using the splitting rule:

t1 |- t2,A&B
-----------------------
t1 |- t2,A t1 |- t2,B

4. eliminate v on the left using the splitting rule:

AvB,S1 |- t2
-----------------------
A,S1 |- t2 B,S1 |- t2

5. t1 |- t2 is a theorem if some formula occurs on both sides of the
turnstile. The initial sequent is a theorem iff all the splits are.

These rules allow sequents containing complex formulas to be decomposed into
sequents containing simpler formulas. The final result is a finite tree of
sequents with the sequents at the leaf nodes containing only atomic formulas.
The sequents at the leaf nodes can be tested for theoremhood by rule 5. If all
the leaf node sequents are theorems then the sequent at the root is a theorem.

Wang's procedure can be viewed as essentially producing DNF on the left and
CNF on the right so that rather than producing a tree and using the decision
procedure just given, one might instead use the following procedure:

translate into negation normal form. & on the left and v on the right
are replaced by commas,

eliminate embedded & on the right and v on the left using distribution
laws to get DNF on left and CNF on the right,

&(Si)|-v(Sj) is a theorem iff each dual clause on the left is
either a contradiction or has a atomic formula in common with each clause on the
right.

RESOLUTION & TABLEAU REVISITED

We offer here an alternative description of Binary resolution in the light
of our discussion of Gentzen systems. The behavior of Binary resolution may be
described by the following algorithm:

i. Given a set of premises t and a conclusion A, then t |- A just in
case t, -A |- , (i.e., t & -A is contradictory), use the Gentzen
rule for negation given above.

We note that the completeness of the resolution rule depends on using CNF on
the left and having the null clause as the consequent.

Conversely, the tableau's behavior may be viewed in this way:

i. given a set of premises t and a conclusion A, then t |- A just in
case t, -A |- , (i.e., t & -A is contradictory). Use the Gentzen
rule for negation given above.

ii. put t, -A in DNF using DeMorgan's law and distributivity.

iii. t|-A iff each dual clause contains a contradiction.

Note that the construction of tableau amounts to putting t, -A in DNF and
that once in DNF on the left of |- determination of theoremhood is
immediate, i.e., just check each dual clause for a contradiction, if there is a
single dual clause that does not contain an atomic formula and its negation
(equivalently, a path that does not clause) then you do not have a theorem and a
valuation witnessing this fact can be read off the dual clause. Note also the
contrast with resolution: resolution starts with CNF and uses the inference rule
to get the empty clause (that is, a clause containing a contradiction) whereas
tableau finish with DNF after which there is nothing to do but read the answer
(at least in the propositional case).

DUALS OF RESOLUTION AND TABLEAU

Each of the Binary resolution and semantic tableau procedures has a dual
procedure. Since both resolution and tableau are refutational (i.e., indirect)
procedures their duals will be non-refutational (i.e., direct) procedures. That
is, while the resolution procedure determines the existence of a valid
consequence relation, t |- , when t is in CNF, the dual of the resolution
procedure determines a valid consequence relation,  |- t, when t is in
DNF. Similarly, whereas a valid consequence relation is found to exist by the
tableau procedure for t |- , when t is in DNF, the dual of the resolution
procedure determines a valid consequence relation,  |- t, when t is in
CNF.

Dualizing the arguments justifying resolution and tableau yields the
non-refutational procedures dual resolution (DNF on the right) and dual tableau
(CNF on the right). The appropriate conversion routines can be shown to be:

dual resolution:

i. put in DNF on right

ii. inference rule: from ... (p&S1)(-p&S2)...

infer ...(S1&S2)(p&S1)(-p&S2)...

iii. condition for determination of theorem:

null dual clause.

dual tableau

i. use inference rules to get CNF on right

ii. theorem just in case each clause contains

pv-p, for some p.

OBSERVATIONS

The characterization of resolution and tableau-based inferencing and their
dual procedures given above is in terms of the normal form representation which
underlies the inferencing and the proof strategy. The distinction between CNF
and DNF and Direct vs. Indirect proof yields the following pairwise comparison:

indirect(LH)

direct(RH)

CNF

Resolution

Dual Tableau

DNF

Tableau

Dual Resolution

Wang is a mixed system (right and left sides).

The most commonly used ATP procedures, resolution and tableau, are indirect
in style (left-sided). We have shown that there are corresponding direct
procedures (right-sided). Wang's algorithm, in contrast, is two-sided since it
incorporates operations for both the left and right hand sides. Analyzing these
procedures in terms of the normal forms used illustrates the similarities and
emphasizes the differences. Each of these procedures are amenable to
automation.

ILLUSTRATION OF THE INFERENCING TECHNIQUES

Consider the following valid argument form

(A) p-(q-r), p-q |- p-r

We now give a proof of this argument in each of the afore-mentioned systems.

resolution

First, we observe that the negation normal form (Horn clause equivalent) of
(A) is:

-pv-qvr, -pvq |- -p, r

which by Gentzen's negation rule:

-pv-qvr, -pvq, p, -r |-

which is the CNF/indirect proof representation of the argument form
consistent with the above classification.

We then proceed by means of Binary resolution:

pv-qvr, -pvq, p, -r |-

pv-qvr, -qvr, -pvq, -r, p |-

pv-qvr, -q, -qvr, -pvq, -r, p |-

pv-qvr, -q, -p, -qvr, -pvq, -r, p |-

pv-qvr, -q, (), -p, -qvr, -pvq, -r, p |-

tableau

Once again, the indirect proof equivalent to (A) in negation normal form is:

-pv-qvr, -pvq, p, -r |-

which converts to the following DNF equivalent

(p&-r&-p&-q) v (p&-r&-p&r) v (p&-r&q&r) |-

since an atomic formula and it's negation appear in each disjunct (path) the
DNF expression is necessarily false (all paths clause).

Wang

Wang's algorithm yields DNF on the left and CNF on the right. Once again,
the negation normal form of (A) is:

-pv-qvr, -pvq |- -p,r

Which is further transformed by Wang's algorithm into the into the following
six DNF sub-arguments

p |- p,r

p,q |- p,r

p |- p,r,q

p,q |- r,q

r,p |- p,r

p,r,q |- r

since in each sub-argument some atom appears on each side of the turnstile
the original argument is a theorem.

In our notation, we create the Gentzen equivalent of a Wang derivation in
this way. First, using distributivity

-pv-qvr, -pvq |- -p,r

is seen to be equivalent to:

-p v (-p&q) v (-q&-p) (r&-p) v (r&q) v F |- -pvr

Which is valid since each non-F (false) dual clause on the left shares an
atomic formula with every clause on the right.

dual of tableau

The direct form of (A) is

|- (p-(q-r))-((p-q)-(p-r))

|- (p&q&-r), (p&-q), -p, r

|- ((-pvp)&(-pvq)&(-pv-r)), (p&-q), r

|- ((rv-pvp)&(rv-pvq)&(rv-pv-r)), (p&-q)

|- (rv-pvp)&(rv-pvqvp)&(pvrv-pv-r)&(-qvrv-pvp)&

(-qvrv-pvq)&(-qvrv-pv-r)

which is in CNF and is thus a theorem since each dual clause contains some
atom and its negation.

dual of resolution

The direct form of (A) in DNF is:

-pv-qvr, -pvq |- -p, r

which by Gentzen negation is equivalent to:

|- p&q&-r, p&-q, -p, r

which is the DNF/direct proof representation of the argument form consistent
with the above classification.

Using the dual resolution rule:

|- p & q & -r, p & -q, r, -p

|- p & q & -r, q & -r, p & -q, r, -p

|- p & q & -r, q, q & -r, p & -q, r, -p

|- p & q & -r, q, p, q & -r, p & -q, r, -p

|- p & q & -r, q, (), p, q & -r, p & -q, r, -p

CONCLUSION

The process of first transforming propositional logic argument forms into
negation normal form and then considering how the standard ATP procedures treat
these forms suggests several procedural extensions. This paper has presented
the two most obvious extensions: dual resolution and dual tableau. These
procedures (a) mirror resolution and tableau and (b) show that ATP need not be
restricted to indirect proof strategies as is the current custom.

Consideration of the NNF of an argument can help guide automated
inferencing. If it could be easily determined that the the conversion of an
argument form into the schema, CNF |- , is less complex than say,  |-
DNF, then the dual resolution procedure would seem more appropriate. Our
present work is motivated by the relationships between the simplicity of the
normal form on the one hand and the most 'natural' inferencing technique on the
other. The present study focuses our attention on this complex relationship
and gives rise to the question of whether an algorithm may be developed which
assigns to each problem an 'optimal' inferencing strategy. This question is
currently being investigated by the authors.

While it cannot be determined at this point whether efficient automated
theorem provers can be developed which will be 'intelligent' enough to select
their own optimal strategies, we are confident that the present taxonomy
provides a useful framework by means of which the logical alternatives may be
compared and contrasted.

REFERENCES:

For a general discussion of the logical aspects of the topics covered above,
consult references [5] or [6].